Scalar Form Factors just beyond the Tree Approximation

Abstract

Using the algebra of currents and current divergences derived from the chiral $\mathrm{SU}(3)$ symmetry-breaking model of Gell-Mann, Oakes, and Renner and a method of pole and cut dominance, we derive two-parameter effective-range formulas for scalar $(I=J=\frac{1}{2})K\ensuremath{\pi}$ and $(I=J=)\ensuremath{\pi}\ensuremath{\pi}$ form factors. The fit to the $I=\frac{1}{2}$ $s$-wave $K\ensuremath{\pi}$ phase-shift data of Antich et al. favors their "lower" solution with ${m}_{K}=1150$ MeV and supports the existence of a broad $\ensuremath{\kappa}$ resonance. It is shown that the introduction of the similarly broad $\ensuremath{\epsilon}$ resonance which emerges from these considerations into the weak-interaction scheme ${{K}_{S}}^{0}\ensuremath{\rightarrow}\ensuremath{\epsilon}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ makes agreement with the experimental value of the ${{K}_{L}}^{0}\ensuremath{-}{{K}_{S}}^{0}$ mass difference difficult, if the ${{K}_{L}}^{0}$ mass shift is not small.